For we say a set $I$ and a binary relation $\leq$ form a pre-ordered set if $\leq$ is reflexive and transitive on $I$.
In a catergory $\mathcal C$, say a family of objects $(C_a)_{a\in I}$ and morphisms $c_{bc} : C_c \rightarrow C_b$ whenever $b \leq c$ such that:
$c_{bc}c_{cd} = c_{bd}$ whenever $b \leq c$ and $c \leq d$ in $I$,
is called an inverse system of shape $(I, \leq )$.
For such an inverse system $C = (C_a, c_{bc})$ we say that the inverse limit $\lim_{\leftarrow_I}C$, if it exists, is an object of $\mathcal C$ which along with morphisms $\pi_a : \lim_{\leftarrow_I}C \rightarrow C_a$ for all $a \in I$ such that: $c_{bc}\pi_c = \pi_b$ whenever $b\leq c$, it satisfies the universal property:
If $D$ is an object of $\mathcal C$ and $\rho_a : D \rightarrow C_a$ for all $a \in I$ satisfying $c_{bc}\rho_c = \rho_b$ whenever $b \leq c$, then:
$\exists! f \in Hom_{\mathcal C}(D, \lim_{\leftarrow_I})$ such that: $\pi_af = \rho_a$ for all $a \in I$.
Now, I am asked to show that in the case that there is a maximal element $t \in I$, and an inverse system $C = (C_a, c_{bc})$ of shape $(I, \leq)$ in a category $\mathcal C$, then $C_t$ along with the morphisms $\pi_a = c_{at}$ are the inverse limit of $C$.
It is easy to see that $C_t$ and these morphisms satisfy $c_{bc}\pi_c = \pi_b$ whenever $b \leq c$, given the definition of the inverse system.
It remains to show the universal property from above, so suppose we have an object $D$ of $\mathcal C$ along with morphisms $\rho_a : D \rightarrow C_a$ for all $a \in I$ such that $c_{bc}\rho_c = \rho_b$ whenever $b \leq c$.
Then note $\rho_t$ is a morphism as required in the universal property, so it remains to show it is unique.
Suppose then that $\exists f \in Hom_{\mathcal C}(D, C_t)$ such that: $\pi_af = \rho_a$ for all $a \in I$, and suppose also that $\exists x \in D$ such that $f(x) \neq \rho_t(x)$. I then need some sort of contradiction.
However, I am not really sure how to arrive at this conclusion?
It is easy to see that we should have:
$c_{tt}f(x) = \rho_t(x) = c_{tt}\rho_t(x)$, so we had some sort of injectivity condition on $c_tt$ then we would be done. However, I cannot immediately see why that would be the case?
I'm not stuck and can only make conclusions along the lines of "if it's not injective then it's not injective". How might I proceed with this question?