Verification of category

Question

We have a category $Pno$. Objects of this category are structures $(A, \alpha, a)$ and arrows are functions $f: (A, \alpha, a) \to (B, \beta, b)$ such as $f \circ \alpha = \beta \circ f$ and $f(a) = b$, i.e. structure-preserving morphisms. I need to verify that $Pno$ is indeed an category . How do I do that? I need to show that there always exists composition of arrows and that this composition is associative, but I don't know how to proceed with this kind of proof.

Answer 1

Given $f\in \hom((A,\alpha,a),(B,\beta,b)),\ g\in\hom((B,\beta,b),\ (C,\gamma,c))$, the first candidate that comes to mind for $gf$ is the set-theoretic function $g\circ f$. You need to verify that $g\circ f\in\hom((A,\alpha,a),\ (C,\gamma,c))$, i.e. that $[g\circ f](a)=c$ and that $(g\circ f)\circ \alpha=\beta\circ(g\circ f)$. Once you've done this, associativity follows from the fact that the usual composition of functions is associative.

In this case, proving that composition preserves commutative properties is quite immediate, since it amounts to using the associativity of function composition twice and the hypothesis