In TOM LEINSTER's Basic Category Theory, Page 25, it says
I got very confused here. Doesn't the definition of injective says for distinct $f_1, f_2$ we have $F(f_1)\neq F(f_2)$? Why this is false? I think I must have some misunderstandings about the definition. Can anyone explain this for me and give some concrete examples?
Let $f_1$ and $f_2$ be distinct morphisms. Then there must be objects $A_1,A_1',A_2,A_2'$ in $\mathcal A$ such that $f_1\in\mathcal A(A_1,A_1')$ and $f_2\in\mathcal A(A_2,A_2')$. Faithfulness of $F$ implies that if $A_1=A_2$ and $A_1'=A_2'$, we must have $F(f_1)\neq F(f_2)$.
But if $A_1\neq A_2$ or $A_1'\neq A_2'$ we can have that $F(f_1)=F(f_2)$ even though $F$ is faithful. For instance let $\mathcal A$ have two objects, $A_1,A_2$ and let the identity morphisms $1_{A_1}$ and $1_{A_2}$ be the only morphisms in $\mathcal A$. Fix an object $B\in\mathcal B$. We define $F:\mathcal A\to\mathcal B$ by $F(A_1)=F(A_2)=B$, and $F(1_{A_1})=1_{B}=F(1_{A_2})$. So we have that $F(1_{A_1})=F(1_{A_2})$ even though $1_{A_1}\neq 1_{A_2}$. Nevertheless, because the homsets $\mathcal A(A_i,A_j)$ for $i,j=1,2$ are either empty or a singleton set, we have that $\mathcal A(A_i,A_j)\to \mathcal B(F(A_i),F(A_j)), f\mapsto F(f)$ is injective, hence $F$ is faithful.