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, however neither the Kleisli bicategory nor the 2-category of free pseudoalgebras are the analogous weighted colimit. 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 the $\mathbf{Gray}$-category of enriched presheaves on $\mathbf{Psmnd}$.