Closure of morphisms with respect to composition.

Question

Let $C$ denote some arbitrary category, and denote $X,Y,Z\in\text{Obj}(C)$. If $\mu_{XY}\in\text{Hom}_C(X,Y)$ and $\mu_{YX}\in\text{Hom}_C(Y,Z)$ are well-defined (total) functions, then can we conclude that $\mu_{YZ}\circ\mu_{XY}\in\text{Hom}_C(X,Z)$? I know by definition of a category that morphisms need not be closed under composition in Mor$(C)$; however, my question pertains to whether this outcome is a result of the fact that some morphisms are not necessarily total functions (hence, such composition would be undefined, proving the lack of closure under composition) or if Mor$(C)$ acts more so as an arbitrary collection of maps lacking this additional property. (I feel I have overthought the situation...but best to check.)

Answer 1

Categories are closed under composition, locally.

In category theory the word "local" or "locally" is used for a property that holds on hom-sets $Hom_C(X,Y)$ for objects $X$ and $Y$.

You may consider for each objects $X,Y,Z$ a map $\_\circ_{X,Y,Z}\_: Hom_C(Y,Z)\times Hom_C(X,Y) → Hom_C(X,Z)$, and this map is indeed total.

It must also satisfy the transitivity and identity properties.

Then $\_\circ\_ : Mor(C) → Mor(C)$ is the "union" $\bigcup_{X,Y,Z}\_\circ_{X,Y,Z}\_$ as a graph if you will.