Functors between morphism categories

Question

Let ${\cal{C}}$ and ${\cal{D}}$ be categories, and let $\textsf{Mor}({\cal{C}})$, $\textsf{Mor}({\cal{D}})$ be their morphism categories (objects are morphisms, morphisms are pairs of morphisms making the appropriate square commute). Let $\textsf{Mor}({\cal{C}})\xrightarrow{F}\textsf{Mor}({\cal{D}})$ be a (covariant) functor, and suppose moreover that it is an isomorphism of categories.

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My question is if $F$'s map on objects necessarily sends identity morphisms of $\cal{C}$ to identity morphisms of ${\cal{D}}$; if it is true, could I perhaps have a hint as to how to approach proving this?

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Here's my background. Obviously an isomorphism of categories ${\cal{C}}\cong{\cal{D}}$ induces an isomorphism of categories $\textsf{Mor}({\cal{C}})\cong\textsf{Mor}({\cal{D}})$, and I was wondering if the converse is true (in fact, I was wondering this about the category of isomorphisms first, but after little progress decided to switch to all morphisms). My gut says that it should be true; in a very loose sense, $\textsf{Mor}({\cal{C}})$ should "fully characterize" ${\cal{C}}$ (and certainly contains an embedded copy of ${\cal{C}}$). The purpose for my question is because I have managed to show if $F$ maps identity morphisms to identity morphisms, then I can construct the (obvious) isomorphism ${\cal{C}}\cong{\cal{D}}$.

Thank you for any insight/help!

(As an aside, does MSE have any native support for a commutative diagram package? I am quite proficient with tikz/tikz-cd but can only use them here by compiling locally, clipping the pdf, and then embedding the clipped image.)

Answer 1

I don't have good intuition for this, so I might be wrong, but I don't see why this should be true: Mor(C) doesn't capture much about the composition in C.

Here's my attempt at a counterexample. Let C = D be the cyclic group {1,g} of order 2, thought of as a category with a single element x with two automorphisms 1 and g. Then, if I understand correctly, Mor(C) = Mor(D) is the following category:

• the elements are 1 and g;
• there are 8 morphisms:
  1. $\mathbf{id}_1 = (1,1): 1\to 1$
  2. $\mathbf{g}_1 = (g,g): 1\to 1$
  3. $\mathbf{id}_g = (1,1): g\to g$
  4. $\mathbf{g}_g = (g,g): g\to g$
  5. $\alpha_{1,g} = (1,g): 1\to g$
  6. $\alpha_{g,1} = (1,g): g\to 1$
  7. $\beta_{1,g} = (g,1): 1\to g$
  8. $\beta_{g,1} = (g,1): g\to 1$.

(It would be much easier if I could draw this, but I'll let you do it.) Now let F be the functor

• swapping the elements 1 and g;
• sending
  1. $\mathbf{id}_1 \leftrightarrow \mathbf{id}_g$
  2. $\mathbf{g}_1 \leftrightarrow \mathbf{g}_g$
  3. $\alpha_{1,g} \leftrightarrow \alpha_{g,1}$
  4. $\beta_{1,g} \leftrightarrow \beta_{g,1}$.

I think F is an isomorphism from Mor(C) to Mor(D), but as it sends 1 to g, it can't come from a morphism from C to D.