Examples of categories where morphisms are not functions

Solution 1:

Consider the category with exactly two objects $a$ and $b$ (they can be anything you want!) and which has exactly one non-identity morphism —going from $a$ to $b$— which is my chair.

Solution 2:

Let $G$ be a directed graph. Then we can think of $G$ as a category whose objects are the vertices in $G$. Given vertices $a, b \in G$ the morphisms from $a$ to $b$ are the set of paths in the graph $G$ from $a$ to $b$ with composition being concatenation of paths. Note that we allow "trivial" paths that start at a given vertex $a$, traverse no edges, and end at the starting vertex $a$. We have to do this for the category to have identities.

Also we can create categories whose morphisms are not exactly maps but instead equivalence classes of maps. For example given an appropriately nice ring $R$ we can create the stable module category whose objects are $R$-modules. Given two $R$-modules $A$ and $B$ the set of morphisms from $A$ to $B$ is the abelian group of $R$-module homomorphisms from $A$ to $B$ modulo the subgroup of homomorphisms that factor through a projective $R$-module.