Is every bijective morphism an isomorphism in a small category? In any category, an isomorphism is an arrow $f \colon X \to Y$ together with an inverse arrow $f^{-1} \colon Y \to X$ such that $f \circ f^{-1} = 1_{Y}$ and $f^{-1} \circ f = 1_{X}$.
Now, clearly in some (algebraic) categories like $\mathsf{Group}$, $\mathsf{Set}$, etc. this definition is equivalent to the bijective morphisms. In others like $\mathsf{Pos}$, there are bijective morphisms between non-isomorphic posets.
But do we have equivalence in all small categories? Since any small category is isomorphic to one where the objects are sets and the arrows are functions.
 A: There is no such thing as a "bijective morphism" in a category. The notion of "bijective" only makes sense in a concrete category, which is a category $C$ equipped with the additional data of a faithful functor $U : C \to \text{Set}$, and the meaning of "bijective" depends on the choice of $U$.
The failure of "bijective" morphisms to be isomorphisms has nothing to do with smallness, and any counterexample you've ever seen can be converted into a small counterexample by ignoring all objects other than the two objects involved in the counterexample.
In fact here is a minimal counterexample: take $C$ to be the category $\{ \bullet \to \bullet \}$ consisting of two objects and a morphism $f$ from one to the other, and take $U : C \to \text{Set}$ to be the functor sending both objects to the one-element set and $f$ to the unique function from this set to itself. Then $U(f)$ is a bijection but $f$ is not an isomorphism because there aren't any morphisms in the other direction at all. Here $C$ is not only small but finite.
