induced morphism between limits and functor

Let \(\mathcal{C}, \mathcal{D}\) be categories and \(\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}\) be a functor. Suppose \(I: \mathcal{I} \rightarrow \mathcal{C}\) is a diagram and

\begin{align*} \mathrm{lim}_I \in& \mathrm{Ob}(\mathcal{C}) \\ \mathrm{lim}_{\mathcal{F} \circ I} \in& \mathrm{Ob}(\mathcal{D}) \\ \end{align*}

the limit

Then there exists a canonical induced morphism

\begin{align*} \mathcal{F}(\mathrm{lim}_{I} \rightarrow \mathrm{lim}_{\mathcal{F} \circ I} \end{align*}

dually to morphism between colimits and functor