Julio wrote:

>\[

\begin{matrix}

& & h & & \\

& c & \rightarrow & c' &\\

f & \downarrow & & \downarrow & g\\

& d & \rightarrow & d' &\\

& & ? & & \\

\end{matrix}

\]

> why must the question mark function be equivalent to some composition of \\(f, h,\\) and \\(g\\)? Can't there simply exist an independent~~function~~ morphism in \\(\mathcal{C}\\) from \\(d\\) to \\(d'\\)?

Good question. But consider the example where the only objects and morphisms in \\(\mathcal{C}\\) are those shown in the picture - and composites of what's shown, and identity morphisms. There's no reason there should be anything else! Then you're stuck.

This is a great example of a general principle in category theory: _you can't make an omelette if you don't have eggs_. You can only cook with the ingredients you have.

You can try to wriggle out of this in various ways, and it would be educational to try.

Stephen explained it another way:

> My CS intuition would say: Anything else that wasn't trivial, would require information we don't have. There aren't really other choices if we want the functor to use the info we have and no more.

To fully get this intuition, I think one has to fight against it for a while and see all the bad things that happen.

>\[

\begin{matrix}

& & h & & \\

& c & \rightarrow & c' &\\

f & \downarrow & & \downarrow & g\\

& d & \rightarrow & d' &\\

& & ? & & \\

\end{matrix}

\]

> why must the question mark function be equivalent to some composition of \\(f, h,\\) and \\(g\\)? Can't there simply exist an independent

Good question. But consider the example where the only objects and morphisms in \\(\mathcal{C}\\) are those shown in the picture - and composites of what's shown, and identity morphisms. There's no reason there should be anything else! Then you're stuck.

This is a great example of a general principle in category theory: _you can't make an omelette if you don't have eggs_. You can only cook with the ingredients you have.

You can try to wriggle out of this in various ways, and it would be educational to try.

Stephen explained it another way:

> My CS intuition would say: Anything else that wasn't trivial, would require information we don't have. There aren't really other choices if we want the functor to use the info we have and no more.

To fully get this intuition, I think one has to fight against it for a while and see all the bad things that happen.