Given is category Set Given two objects from this category, $A$, and $B$, which are sets without any other structure, there is an arrow $f: A \to B$, from $A$ to $B$, which is any total function from $A$ to $B$.  For given set $A$, there is ( by axiom of CT ) identity arrow $1_A$ .  Then it looks like identity arrow may be a permutation...  Buy identity arrow for a given object has to be unique. And there is many permutations for given set $A$.   So do we have here "equivalence class" of identity arrow?
I've been trying to wrap my head around the basic concepts of category theory, and I thought I would attempt to illustrate what I understand with the category of sets, probably the easiest example. Particularly, I've been trying to prove that $id_A$ (the identity morphism on $A$, for all $A \in Obj(\mathbf{Set})$) is $1_A \colon A \rightarrow A, x \mapsto x$.  This is a very intuitive and reasonable statement, and it's trivial to prove that $1_A$ is indeed an identity morphism on $A$, and I suppose uniqueness of $id_A$ can be demonstrated analogously to uniqueness of the identity element in a monoid (considering the subcategory which has $A$ as its only object, and endofunctions on $A$ as its only morphisms).  In this manner, it is not hard to prove that the proposition in the title is true, but this demonstration requires to make an assumption or guess as to what could $id_A$ be. Specifically, the scheme of the proof is: assume $id_A$ = $1_A$, see that it works with the definition of an identity morphism, show that the identity morphism is unique, and, in conclusion, $id_A$ can only be $1_A$. What I'm looking for, nonetheless, is a somehow more direct proof, that doesn't assume $id_A$ = $1_A$ at the start. I want to place myself in a state of little or no knowledge about sets and functions, and under this assumption, why would I assume $id_A$ = $1_A$ at first? Why not try with $id_\mathbb{Z}$ = $f \colon \mathbb{Z} \to \mathbb{Z}, x \mapsto x^2 + 1$, for example? It wouldn't work, but I don't have any reason to think that $1_\mathbb{Z}$ is a better guess for $id_\mathbb{Z}$.  I suppose that the proof for which I'm asking would work for categories of sets with additional structure, and probably for posets as well, although I'm not clear as to what modifications it would require to work.  Thanks.
My Linux Mint 15 install on VirtualBox was working great up until about two days ago, when the updater started throwing a ton of errors and refuses to progress with any apt-get update, apt-get install, or apt-get upgrade.  They are 404 errors on the archive.ubuntu.com site, under the /ubuntu/dists/raring-updates folder. I looked at it, it doesn't exist.  a direct link, which shows there is no raring-updates.  Where has raring-updates gone and what can I do to fix this? I would like to get the multi pattern match with implicit AND between patterns, i.e. equivalent to running several greps in a sequence:  grep pattern1 | grep pattern2 | ...   So how to convert it to something like?  grep pattern1 &amp; pattern2 &amp; pattern3   I would like to use single grep because I am building arguments dynamically, so everything has to fit in one string. Using filter is system feature, not grep, so it is not an argument for it.    Don't confuse this question with:  grep "pattern1\|pattern2\|..."   This is an OR multi pattern match. Can I Create a Dictionary of Generic Types? Area 51 Site is quite different from other SE sites and needs an SE footer Honestly, navigation through Area 51 is a bit confusing and seems to be much different from the other SE sites.    The feature that is very lacking is the footer with all the sites you are subscribed to so that a user can easily return to a different SE site. This one really bugs me.  Can this be easily fixed?  Also, unlike all the other sites there is no "meta" site to add comments specifically about that site.    Again Area 51 seems out of sync with the other SE sites. Gambler's ruin and coin toss I'm using classicthesis to write my document. I don't use the option drafting but I can see on the bottom of the page the date (there is a part of the date that is not readable, but when I copy and paste this part I obtain [ December 1, 2011 at 20:50 ]). This is my code:  \documentclass[a4paper]{scrreprt}   % classe report di KOMA-Script  \usepackage[english]{babel}     % Per scrivere in inglese \usepackage[T1]{fontenc}        % Imposta la codifica dei font \usepackage[utf8]{inputenc}     % Lettere accentate da tastiera \usepackage{classicthesis-preamble} % Preambolo classicthesis \usepackage{classicthesis}      % Classicthesis  \begin{document}  \input{Chapters/0_Introduction} [...]  \end{document}   I would like to avoid this little issue.  I use TeXworks in Windows7 x64. How many inventions did Fred and George have? Fred and George Weasley are known for their trouble-making and general &quot;solemnly swearing they are up to no good&quot;. Their antics/adventures give something to smile about and show us not all wizards are as serious as Professor McGonagall and Hermione. This gives them a reputation of being less intelligent and indeed they only achieved three OWLs. However, we also see the twins have quite a brilliant side! Being able to open and successfully run a shop in Diagon Alley without having completed their Hogwarts education is no easy take (IMO, anyway). Their creations helped students get out of class, our heroes out of tight spot, and even stop our enemies execute an evil plot. Ginny may have said it best:  &quot;We’ve been hearing explosions out of their room for ages, but we never thought they were actually making things,&quot; said Ginny. &quot;We thought they just liked the noise.&quot; Harry Potter and the Goblet of Fire  Do we know how many things Fred and George invented? space on disk running low error logging /var/log/cups/error.log I have Lubuntu 16.10 installed on my disk mounted as /.  The problem is that the CUPS log file( /var/log/cups/error.log)  grows all the time until I no have free space left on the disk...   So when I delete that file, the space on disk freed up again  I already tried:    fsck disk usage analyzer every apt-cleaning/autocleaning, and dependencies fixing.    How should I proceed?  du -sxh:  9,0G    .  Model: ATA ST3160815AS (scsi) Disk /dev/sda: 160GB Sector size (logical/physical): 512B/512B Partition Table: msdos Disk Flags:   Number  Start   End     Size    Type      File system     Flags  1      32,3kB  50,3GB  50,3GB  primary   ext4            boot  2      50,3GB  160GB   110GB   extended  5      50,3GB  158GB   108GB   logical   ext4  6      158GB   160GB   2136MB  logical   linux-swap(v1)   parted --list &amp;&amp; sudo df -h:  Model: ATA ST3160815AS (scsi) Disk /dev/sda: 160GB Sector size (logical/physical): 512B/512B Partition Table: msdos Disk Flags:   Number  Start   End     Size    Type     File system  Flags  1      32,3kB  50,3GB  50,3GB  primary  ext4         boot  2      50,3GB  160GB   110GB   primary  ext4   Filesystem      Size  Used Avail Use% Mounted on udev            985M     0  985M   0% /dev tmpfs           201M  6,3M  195M   4% /run /dev/sda1        46G   43G  709M  99% / tmpfs          1003M  188K 1003M   1% /dev/shm tmpfs           5,0M  4,0K  5,0M   1% /run/lock tmpfs          1003M     0 1003M   0% /sys/fs/cgroup tmpfs           201M   36K  201M   1% /run/user/1000 /dev/sda2       101G   60M   96G   1% /media/aram/ I'll be arriving at Heathrow in June 2015 with my current girlfriend for a holiday in the UK. I've been trying to find out which immigration queue to join at Heathrow Passport Control without much luck. I have a British passport, my girlfriend has a non-EU passport.     Should I join the EU queue and she the non-EU queue or should we both queue together (and if so, which of the queues should we join)? How do I handle partially correct edits in the Suggested Edits review queue? Contradicting Instructions While going through the suggested edits review queue, one sometimes stumbles upon partially useful edits like . Trying to follow the instructions listed in the descriptions doesn't lead to a conclusive answer. Considering our example:   Approve edits that clearly improve the post ↳ No, not *clearly* Improve Edit when you can make additional improvements to the post ↳ I most definitively can Reject and Edit to replace an ineffective edit with your own substantive changes ↳ It is ineffective in the sense that it made the post worse overall (imho) Reject edits that fail to improve the post or that make it worse ↳ Like the last option: yes Skip if you are not sure and want to go to the next suggested edit ↳ Well, yes. But that is really why I am here, asking this question   Reasoning about the remaining Choices This leaves us with three choices: Improve Edit, Reject and Edit and Reject (the last one slightly modified):  Improve Edit  + Acknowledges that some things in the edit where correct - Learning effect for author might vanish, as the improvement might get overlooked - Can't leave a custom message for edit author (except in comments, which pollutes the thread) - Doesn't trigger the system which &quot;punishes&quot; users who repeatedly have their edits rejected - Rewards the edit author with reputation for a poor edit   Reject and Edit  + Clearly shows the author that there were problems with the edit - Still can't leave a custom message explaining the problems of the original post. - One has to partially redo parts of the edit that have just been rejected, which might feel unjust to the original editor   Reject with a custom note and come back immediately to make a good Edit  + Clearly shows the author that there were problems with the edit + Possibility to leave an explanation as to what was wrong with the edit - One still has to redo parts of the edit that has just been rejected - Seems to be a weird workaround for Reject and Edit   Something else?   Each of the three options come with their distinct advantages/disadvantages making it non trivial to decide which one to pick.  Or does the answer simply depend on the quality of the edit? Like:  Improve Edit for subjectively slightly positive edits Reject and Edit (Or Reject and Edit) for subjectively slightly negative edits How do I delete Windows after an Ubuntu install? I downloaded Ubuntu 12.04 LTS to my Dell Dimension 4600 with Windows XP installed.  I now have a neat dual-boot system, but I want to get rid of the XP OS and keep Ubuntu as my only OS.  How do I do this? Bogus float inequality due to compiler optimization? How to stop fluid inflow? I have fluid simulation with an inflow object. I want the inflow to go for 2 seconds, then have the inflow stop (so no more fluid is added) but keep the simulation going  (so the existing fluid sloshes around).  However, I do not see any way to set start and ending values on my inflow object.  How do I stop inflow at a certain frame/time in my simulation? How to change the desktop wallpaper to a photo of my own selection in Ubuntu 18.04? I'm not interested in downloading some wallpaper changer application, nor am I interested in coding something up. There has to be an easy way to select a photo as a custom wallpaper in Ubuntu 18.04. A few months ago I was introduced to the new  type and was glad 's flaws with regard to time zones were finally taken care of.  However, I was left wondering if there were any overhead or problems that could occur from using this new type.  I work on a multi-locale web application. Does anyone know of anything that could sway me from just using it for all my date/time work? Is there a window for abuse here?  Reference: Help with this query I have a person table. I also have a car table and books table.  The car and book tables both have a foreign key back to person.  I am trying to count the number of books and cars each person has. 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If perimeter of a triangle is $2d$, what is the length of sides so the triangle has maximal area?    I found some solution using circle and angles, but I think I have to use derivatives.    I need help. Why do electrons occupy the space around nuclei, and not collide with them? We all learn in grade school that electrons are negatively-charged particles that inhabit the space around the nucleus of an atom, that protons are positively-charged and are embedded within the nucleus along with neutrons, which have no charge. I have read a little about electron orbitals and some of the quantum mechanics behind why electrons only occupy certain energy levels. However...  How does the electromagnetic force work in maintaining the positions of the electrons? Since positive and negative charges attract each other, why is it that the electrons don't collide with the protons in the nucleus? Are there ever instances where electrons and protons do collide, and, if so, what occurs?