A product is a limit of a diagram indexed by a discrete category, with only identity morphisms. A diagram in $\mathsf{C}$ indexed by a discrete category $J$ is simply a collection of objects $F_j \in \mathsf{C}$ indexed by $j \in J$. A cone over this diagram is a $J$-indexed family of morphisms $(\lambda_j \colon c \to F_j){j \in J}$, subject to no further constraints. The limit is typically denoted by $\prod{j \in J} F_j $ and the legs of the limit cone are maps \(\left(\pi_k \colon \prod_{j \in J} F_j \to F_k\right)_{k \in J}\) called (product) projections. The universal property asserts that composition with the product projections defines a natural isomorphism\n\(\xymatrix{ \cC(c,\prod_{j \in J} F_j ) \ar[r]^-{(\pi_k)_*}_-{\cong} & \prod_{k \in J} \mathsf{C}(c,F_k) \cong \textup{fun}(c,F).}\)