Yoneda Lemma says that, for a locally small category $\mathcal{C}$, an object $A$ in $\mathcal{C}$ and a functor $F:\mathcal{C}^{op}\to \textbf{Set}$, the natural transformations $[Hom(-,A):F]$ is in one-to-one correspondence with elements of $F(A)$.   My question is the following:   $\textit{A priori}$, we do not know that the collection $[Hom(-,A):F]$ is.  Now we show a "bijection" to $F(A)$.  How does it imply that the collection $[Hom(-,A):F]$ is a set.    If the above question does not make sense, then I try to frame it in another fashion.    For doing mathematics we need to consider primitive undefined objects and their "properties".  Mathematicians argue that foundations can be built in many ways like-   sets and $\in$ (Well, wasn't this the plan) sets and $f$ (function) (Leinster's $\textit{Basic Category Theory}$) types (friend told this to me)    So what are the objects we are considering in Category theory. If we take an arbitrary collection of morphisms, when can we say it is a set or not.  If it is not a set, how is its properties different from that of a set.   If this is vague question, specfically considering Yoneda Lemma, how can we say that $[Hom(-,A):F]$ is a set.
Appearing on the second page (under the section Digression: Size worries) of the following paper about the Yoneda Lemma:     It says that $a$ $priori$ $[\mathcal C^{op},Set](H_A,X))$ is a class. I don't understand why this is the case.  My understanding is that $a$ $priori$ each member of $[\mathcal C^{op},Set](H_A,X))$ is a class.  So it seems to me that $[\mathcal C^{op},Set](H_A,X))$ could be a collection of proper classes.  I've looked at this , but it hasn't helped me much.  Any help is appreciated -Thanks
Create instance of generic type in Java? How to remove all white spaces just between brackets [] using Unix tools? Consequence of the Hahn-Banach Theorem From wikipedia's page on the Hahn-Banach Theorem:  If $V$ is a normed vector space with linear subspace $U$ (not necessarily closed) and if $z$ is an element of $V$ not in the closure of $U$, then there exists a continuous linear map $ψ : V → K$ with $ψ(x) = 0$ for all $x$ in $U$, $ψ(z) = 1$, and $||ψ|| = dist(z, U)^{−1}.$  How is this a consequence of the Hahn-Banach Theorem? How much effect does the Bernoulli effect have on lift? I understand that the Bernoulli effect is a flawed explanation for the cause of lift, and does not cause much at all, but how much?   Is there any experimental data on the force caused by the Bernoulli effect? Maybe implicitly through data of the pressure difference between the top and underside of an aeroplane's wings. After that, I assume I could (crudely approximating the pressure to be acting perpendicularly to the flight direction) use $\Delta P A$ to work out the net force on the plane.    Perhaps there is another way to quantitatively analyse the extent to which the Bernoulli effect causes lift.  Edit: see  short cartoon (content similar to Mike Dunlavey's answer). How can I override the OnBeforeUnload dialog and replace it with my own? Animagi (Sirus specifically) and wands? When you pick up items in Skyrim it tells you the value of each item. However when I go and sell the items the merchants are only paying 30% of this value, at best.  Discounting any perks or other effects, how high does Speechcraft have to be before merchants start paying full value for items? Working with IPv6 Addresses in PHP After searching around somewhat thoroughly, I noticed a slight lack of functions in PHP for handling . For my own personal satisfaction I created a few functions to help the transition.  The IPv6ToLong() function is a temporary solution to that brought up here: . It will split the IP in to two integers and return them in an array.  /**  * Convert an IPv4 address to IPv6  *  * @param string IP Address in dot notation (192.168.1.100)  * @return string IPv6 formatted address or false if invalid input  */ function IPv4To6($Ip) {     static $Mask = '::ffff:'; // This tells IPv6 it has an IPv4 address     $IPv6 = (strpos($Ip, '::') === 0);     $IPv4 = (strpos($Ip, '.') &gt; 0);      if (!$IPv4 &amp;&amp; !$IPv6) return false;     if ($IPv6 &amp;&amp; $IPv4) $Ip = substr($Ip, strrpos($Ip, ':')+1); // Strip IPv4 Compatibility notation     elseif (!$IPv4) return $Ip; // Seems to be IPv6 already?     $Ip = array_pad(explode('.', $Ip), 4, 0);     if (count($Ip) &gt; 4) return false;     for ($i = 0; $i &lt; 4; $i++) if ($Ip[$i] &gt; 255) return false;      $Part7 = base_convert(($Ip[0] * 256) + $Ip[1], 10, 16);     $Part8 = base_convert(($Ip[2] * 256) + $Ip[3], 10, 16);     return $Mask.$Part7.':'.$Part8; }  /**  * Replace '::' with appropriate number of ':0'  */ function ExpandIPv6Notation($Ip) {     if (strpos($Ip, '::') !== false)         $Ip = str_replace('::', str_repeat(':0', 8 - substr_count($Ip, ':')).':', $Ip);     if (strpos($Ip, ':') === 0) $Ip = '0'.$Ip;     return $Ip; }  /**  * Convert IPv6 address to an integer  *  * Optionally split in to two parts.  *  * @see https://stackoverflow.com/questions/420680/  */ function IPv6ToLong($Ip, $DatabaseParts= 2) {     $Ip = ExpandIPv6Notation($Ip);     $Parts = explode(':', $Ip);     $Ip = array('', '');     for ($i = 0; $i &lt; 4; $i++) $Ip[0] .= str_pad(base_convert($Parts[$i], 16, 2), 16, 0, STR_PAD_LEFT);     for ($i = 4; $i &lt; 8; $i++) $Ip[1] .= str_pad(base_convert($Parts[$i], 16, 2), 16, 0, STR_PAD_LEFT);      if ($DatabaseParts == 2)             return array(base_convert($Ip[0], 2, 10), base_convert($Ip[1], 2, 10));     else    return base_convert($Ip[0], 2, 10) + base_convert($Ip[1], 2, 10); }   For these functions I typically implement them by calling this function first:  /**  * Attempt to find the client's IP Address  *  * @param bool Should the IP be converted using ip2long?  * @return string|long The IP Address  */ function GetRealRemoteIp($ForDatabase= false, $DatabaseParts= 2) {     $Ip = '0.0.0.0';     if (isset($_SERVER['HTTP_CLIENT_IP']) &amp;&amp; $_SERVER['HTTP_CLIENT_IP'] != '')         $Ip = $_SERVER['HTTP_CLIENT_IP'];     elseif (isset($_SERVER['HTTP_X_FORWARDED_FOR']) &amp;&amp; $_SERVER['HTTP_X_FORWARDED_FOR'] != '')         $Ip = $_SERVER['HTTP_X_FORWARDED_FOR'];     elseif (isset($_SERVER['REMOTE_ADDR']) &amp;&amp; $_SERVER['REMOTE_ADDR'] != '')         $Ip = $_SERVER['REMOTE_ADDR'];     if (($CommaPos = strpos($Ip, ',')) &gt; 0)         $Ip = substr($Ip, 0, ($CommaPos - 1));      $Ip = IPv4To6($Ip);     return ($ForDatabase ? IPv6ToLong($Ip, $DatabaseParts) : $Ip); }   Someone please tell me if I'm reinventing the wheel here or I've done something wrong.  This implementation converts IPv4 to IPv6. Any IPv6 address it doesn't touch. Google Chrome is my favorite browser, and I can't imagine going on without it. Suddenly (without closing and re-opening) the browser; chrome has been reading characters upside down, very weird:    Chrome version:    I tried restarting my browser and my MAC (running OS X Yosemite 10.10), but the error keeps on coming. Is there a Google Drive client available? I'd like to install a  client for Xubuntu (12.04).  I'm getting non-English Google results, and I didn't find  in the Xubuntu repositories.   Does anyone know of a client that works well for Ubuntu or it's supported derivates (or installation instructions for grive?) Several responses to a question  suggested I try Opera kiosk mode, so I decided to download it and give it a try. Once the download had completed and the disk image was being verified, Safari hung for over a minute and there were periods where Safari used a very high %CPU:    In Mac OS X 10.5, I was accustomed to seeing Safari freeze, but it never used up more than 200% CPU (2 cores x 100%). Is this a bug in Activity Monitor? Or does this mean that %CPU is now more akin to ? What is the word for when someone is overly nice and actually isn't nice at all because of the unnecessary added information/comments? For example, they will say things like "Your hair looks so good that wayyyy omg you should do it that way ALL the time! Looks way better than how you usually do it." And the word I'm looking for is not passive aggressive and it is not sarcasm. There is a word for these types of people. What is it? These people think they're being kind but actually blunt and rude. Another example would be for someone to say "hey you're beautiful, but maybe if you didn't wear harsh makeup you would look better" and they intend to sound like they're being nice when you know they don't like you and are saying in out of spite. How do I get my C# program to sleep for 50 milliseconds?  This might seem an easy question, but I'm having a temporary brain failure moment! I am having trouble understanding the estimation of an AR process. In some textbooks, the AR(1) process is defined as follows: $y_{t}=\theta y_{t-1}+ϵ_t$ (which does not contain a constant). So the OLS estimator is biased. I am confused about the cause of the bias. It is explained that $y_{t-1}$ is dependent on $ϵ_{t-1}$ although it is independent of $ϵ_t$. However in linear regression, if the equation does not contain a constant, we cannot make sure the expectation of disturbance is zero. So the OLS estimator is bias without a constant. Does it mean that the OLS estimator is unbiased if I add a constant in AR process? What do the symbols in service --status-all mean? I really miss the graphical "Services" tool from older versions of Ubuntu, but I understand why it had to go. I always have terminals open anyway, so I use (because I have to now, I guess) service --status-all to see what services are running. This would be ok except that the command produces output like the following:   [ + ]  winbind  [ ? ]  wpa-ifupdown  [ - ]  x11-common   I'm guessing that those symbols mean something like listening, blocked, or stopped, but which is which? More importantly, why doesn't the man page say? Apache error "Could not reliably determine the server's fully qualified domain name" When I restart my Apache server using the command  sudo /etc/init.d/apache2 restart   I get the following error:  Restarting web server apache2 apache2: Could not reliably determine the server's fully qualified domain name, using 127.0.1.1 for ServerName ... waiting apache2: Could not reliably determine the server's fully qualified domain name, using 127.0.1.1 for ServerName   Is the server using 127.0.1.1 instead of 127.0.0.1? What causes this error? I was playing my PS4 two days ago and then packed it up with two controllers that were both connected to this PS4.   I got to where I was going, set everything up and now the controllers won't connect. They show pulsing orange when they plug in, but when I hit the PS Button to connect, it blinks white 1-2 times and goes back to orange.   I have tried resetting them both.  Can anyone help? "did shoot" vs "shot" Can we build capacitors on a PCB board? You may have encountered a situation in which Mac changes the order of desktops based on some events like alerts on a program or a web page in browser. How could I disable this logical change? Note: I'm using El Capitan, but I had this problem since Mavericks.