  For a (possibly large) realized limit sketch $\mathcal{S}$ such that every
$\mathcal{S}$-model is small in a suitable sense we show that the category of
cocontinuous functors $\mathsf{Mod}(\mathcal{S}) \to \mathcal{C}$ into a
cocomplete category $\mathcal{C}$ is equivalent to the category
$\mathsf{Mod}_{\mathcal{C}}(\mathcal{S}^{\mathrm{op}})$ of $\mathcal{C}$-valued
$\mathcal{S}^{\mathrm{op}}$-models. From this result we deduce universal
properties of several examples of cocomplete categories appearing in practice.
It can be applied in particular to infinitary Lawvere theories, generalizing
the well-known case of finitary Lawvere theories. We also look at a large limit
sketch that models $\mathsf{Top}$, study the corresponding notion of an
internal net-based topological space object, and deduce from our main result
that cocontinuous functors $\mathsf{Top} \to \mathcal{C}$ into a cocomplete
category $\mathcal{C}$ correspond to net-based cotopological space objects
internal to $\mathcal{C}$. Finally, we describe a limit sketch that models
$\mathsf{Top}^{\mathrm{op}}$ and deduce from our main result that continuous
functors $\mathsf{Top} \to \mathcal{C}$ into a complete category $\mathcal{C}$
correspond to frame-based topological space objects internal to $\mathcal{C}$.
Thus, we characterize $\mathsf{Top}$ both as a cocomplete and as a complete
category. Thereby we get two new conceptual proofs of Isbell's classification
of cocontinuous functors $\mathsf{Top} \to \mathsf{Top}$ in terms of
topological topologies.
