Suppose you have a morphism $f:\mathcal{C}\to \mathcal{D}\times\mathcal{E}$ of categories. Then $f=\langle \pi_{\mathcal{D}}\circ f,\pi_{\mathcal{E}}\circ f\rangle$, uniquely, by the definition of the operation $\langle -,-\rangle$.
If $c\in\text{ob}\;\mathcal{C}$, can we say something about $f(c)$, in terms of this "decomposition"? (Suppose that we know how the objects behave under $\pi_{\mathcal{D}}\circ f$ and $\pi_{\mathcal{E}}\circ f$.)