Let $(X_i,f_{ij})$ be an inverse system of objects and morphisms (say over $\mathbb{N}$) in a category $C$ and let $X$ in $C$ together with morphisms $\pi_i:X\rightarrow X_i$ be the inverse limit of $(X_i,f_{ij})$. Now, let $(Y, \gamma_i)$ be the pair consisting of $Y$ an object in $C$ together with morphisms $\gamma_i$ s.t. $\gamma_i=f_{ij}\circ\gamma_j$ for all $i \leq j$. Then, by definition of the inverse limit, there exists a unique morphism $u:Y\rightarrow X$ s.t. $\gamma_i = \pi_i \circ u$ for all $i$.
Question: is $u$ always injective? Suppose $x \in ker(u)$ and that $x \neq 0$. Then since $\pi_i$ is a morphism and since $\gamma_i = \pi_i \circ u$ we must have that $\pi_i(x)=0$ for all $i$. But is that only possible if $x=0$, giving me a contradiction?