Let $\mathsf{nCob}$ be the category of $n$-cobordisms, whose objects are $(n-1)$-dimensional closed manifolds and morphisms are bordisms. Is this category bicomplete, or even finitely bicomplete?
Once again, I know this is a very simple question, but I couldn't find a single reference on it.
Here is a counterexample for finite (co)completeness when $n = 2$. Consider the two cobordisms $\alpha, \beta : S^1 \to S^1$ given respectively by a cylinder and a cylinder with a hole. Then the equalizer of $\alpha$ and $\beta$ does not exist, because there doesn't even exist a cobordism $\gamma : C \to S^1$ (for some compact 1-manifold $C$) such that $\alpha \circ \gamma = \beta \circ \gamma$: compute its Euler characteristic to reach a contradiction. Similarly the coequalizer doesn't exist. You can adapt the counterexample to any $n \ge 2$ – you just need to find two cobordisms $X \to Y$ with differing Euler characteristic for some $X$ and $Y$.