  A pseudomonad on a $2$-category whose underlying endomorphism is a
$2$-functor can be seen as a diagram $\mathbf{Psmnd} \rightarrow \mathbf{Gray}$
for which weighted limits and colimits can be considered. The $2$-category of
pseudoalgebras, pseudomorphisms and $2$-cells is such a
$\mathbf{Gray}$-enriched weighted limit \cite{Coherent Approach to
Pseudomonads}, however neither the Kleisli bicategory nor the $2$-category of
free pseudoalgebras are the analogous weighted colimit \cite{Formal Theory of
Pseudomonads}. In this paper we describe the actual weighted colimit via a
presentation, and show that the comparison $2$-functor induced by any other
pseudoadjunction splitting the original pseudomonad is bi-fully faithful. As a
consequence, we see that biessential surjectivity on objects characterises left
pseudoadjoints whose codomains have an `up to biequivalence' version of the
universal property for Kleisli objects. This motivates a homotopical study of
Kleisli objects for pseudomonads, and to this end we show that the weight for
Kleisli objects is cofibrant in the projective model structure on
$[\mathbf{Psmnd}^\text{op}, \mathbf{Gray}]$.
