Pullback of coproduct inclusions

Question

Let $X$, $Y$ and $Z$ be objects of a category. Consider the coproduct inclusions $X+Y\to X+Y+Z$ and $Y+Z\to X+Y+Z$. Is it always true that their pullback is $Y$?

I cannot find a counterexample.

Answer 1

The answer is no. The counter-example is a commutative square of distinct objects $W\rightrightarrows X+Y,Y+Z\rightrightarrows X+Y+Z$ together with an additional object $Y$ having a unique morphism to every other object.

In that case, $Y$ is an initial object, which implies that the inclusions $X\to X+Y$, $Z\to Y+Z$, $X+Z\to X+Y+Z$ are isomorphisms, so we may take them to be identities. Moreover, since $X=X+Y$ and $Z=Y+Z$ admit a pair of morphisms only to $X+Z=X+Y+Z$, the latter is indeed a coproduct.

But the pullback of $X\to X+Z\leftarrow Z$ is then $X\leftarrow W\to Z$ since the only morphisms into $X,Z$ are from $Y$ and $W$, correspoding to the unique morphism $Y\to W$ and the identity morphism $W\to W$.