Determining final and initial object in a certain category

Question

I am reading Paolo Aluffi's greatly entertaining book "Algebra: Chapter $0$" and I got stuck on some excercises dealing with universal properties.

Let $C$ be a category, and let $A$ and $B$ be two objects in $C$.

Define $C_{A,B}$ to be the category whose objects are morphisms in $C$ with targets $A$ and $B$ and whose morphisms are commutative diagrams in $C$: \begin{equation} Obj(C_{A,B}):=(Z,f,g)\qquad\text{with $Z\in Obj(C), f\in Hom_C(Z\rightarrow A), g\in Hom_C(Z\rightarrow B)$}\\ \sigma\in Hom_{C_{A,B}}((Z_1,f_1,g_1),(Z_2,f_2,g_2))\Longleftrightarrow f_1=f_2\circ\sigma\quad\text{and}\quad g_1=g_2\circ\sigma \end{equation}

In a similar way, define $C^{A,B}$ to be the category whose objects are morphisms in $C$ with sources $A$ and $B$ and whose morphisms are commutative diagrams in $C$: \begin{equation} Obj(C^{A,B}):=(Z,f,g)\qquad\text{with $Z\in Obj(C), f\in Hom_C(A\rightarrow Z), g\in Hom_C(B\rightarrow Z)$}\\ \sigma\in Hom_{C^{A,B}}((Z_1,f_1,g_1),(Z_2,f_2,g_2))\Longleftrightarrow f_2=\sigma\circ f_1\quad\text{and}\quad g_2=\sigma\circ g_1 \end{equation}

In both cases, $\sigma\in Hom_C(Z_1,Z_2)$.

I understand that $C$ has products if there exists a final object in $C_{A,B}$ denoted $A\times B$, and that $C$ has coproducts if there exists an initial object in $C^{A,B}$ denoted $A\amalg B$ (if $C=$ Set, $A\times B$ is the usual product of sets, and $A\amalg B$ is the disjoint union).

My question is: are there initial objects in $C_{A,B}$ and final objects in $C^{A,B}$? I am completely clueless on where to begin my search. Hints will be greatly appreciated!

Answer 1

To clarify one mistake in your question: $C_{A,B}$ need not even have a final object-among sets this is the usual cartesian product, so if you take for instance some subcategory of sets not closed under cartesian products then there will be no final object.

You normally should not expect such diagrams to be significant. They exist, for instance, over sets, but they're all just the empty set with its empty maps to $A$ and $B$. The issue is that a $(Z,f,g)$ which factors into every $(Z',f',g')$ must be extremely small, heuristically, since $Z'$ can be arbitrarily small, so small that in reasonable cases it just becomes an initial object from $C$. I can't think off the top of my head of a non-artificial example of anything more interesting. The other question is dual, so equivalent.

Answer 2

The answer is: not necessarily. For example, let $C$ be a category with a final object $P$, and let $A = B = P$. Then $C_{A,B} = C$. Therefore $C_{A,B}$ will have an initial object if and only if $C$ does. For example, if $C$ is the category opposite to the category of nonzero rings, then $C$ has a final object $\mathbf{Z}$ but no initial one.