object-injective functor

Let \(\mathcal{C}, \mathcal{D}\) be categories and \(\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}\) be a functor. Then \(\mathcal{F}\) is said to be object-injective, if for each \(A, B \in \mathrm{Ob}(\mathcal{C})\) with \(\mathcal{F}(A) = \mathcal{F}(B)\), it follows, that

\begin{align*} A = B \end{align*}