Enriched Kleisli objects for pseudomonads

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 $[\mathbf{Psmnd}^\text{op}, \mathbf{Gray}]$.