I am reading Rotman's book on Homological algbra and have a slightly different proof of the statement in the title of this question. Am writing my attempt below. Could someone please advise me if I am correct or not ? Thanks. $\newcommand{\mb}[1]{\mathbb{#1}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\dlim}{\lim\limits_{\rightarrow}}$
Given : $\mc{C}, \mc{D}$ are categories. $ F : \mc{C} \to \mc {D} $ and $G: \mc{D} \to \mc{C}$ are functors and $(F,G)$ is an adjoint pair.
To Show : $F$ preserves direct limits.
Proof : Let $\{ C_{i}, \phi^{i}_{j} \}_{i\leq j} $ be a direct system of objects in $\mc{C}$ and $\lim\limits_{\rightarrow} C_i$ be the direct limit. \ Then $ \{ FC_i, F\phi^{i}_{j}\}_{i\leq j} $ is a direct system in $\mc{D}$ and we wish to show that $ F\dlim C_i$ is the direct limit of this system.
On applying $F$ we get the commutative diagram
Suppose $ D $ is an object of $\mc{D}$ and $ \chi_i : FC_i \rightarrow D $ are morphisms such that the diagram
commutes
Then we wish to show that there is a unique map $ \gamma : F \dlim C_i \rightarrow D $ which makes the diagram
commute.
We have available the natural bijections $\tau_{C_i,D}: Hom(FC_i,D) \rightarrow Hom(C_i,GD)$. Let $\psi_i = \tau_{C_i,D}(\chi_i)$ for all $i$. Notice that $\chi_i=\chi_j F\phi^{i}_{j} \implies \psi_i=\psi_j \phi^{i}_{j}$.Then the diagram
is commutative where the morphism $\beta : \dlim C_i \rightarrow GD $ is obtained by using the universal property of direct limit. The map $ \gamma = (\tau_{\dlim C_i,GD})^{-1}(\beta) : F\dlim C_i \rightarrow D $ satisfies the required commutation relations and is the desired map. The uniqueness follows from the uniqueness of the map $\beta$. Hence $F \dlim C_i$ is the direct limit.
Yes, it seems perfect.
(Note that you never used that the given diagram is a directed system.)