Enriched morphisms

Construction 19.4.2 (cf. [Lurie (2017), Definition 4.7.1.1]). Let \(p_D\colon D^{\otimes } \to \oLMod ^{\otimes }\) exhibit \(D\) as left-tensored over \(C\). An enriched morphism of \(D\) is a diagram \[ N \xleftarrow {\beta } X \xrightarrow {\alpha } M \] in \(D^{\otimes }\) satisfying the following conditions:

(1)

The morphism \(\beta \) is \(p_D\)-cocartesian and the image \(p_D(\beta )\) in \(\oLMod ^{\otimes }\) is the projection map \(\pr \colon (\{\fa \},\{\fm \}) \to \{\fm \}\);

(2)

The image \(p_D(\alpha )\) in \(\oLMod ^{\otimes }\) is the action map \(\act \colon (\{\fa \},\{\fm \}) \to \{\fm \}\).

We denote by \[ \Ar ^{\enr }(D) \subseteq \Fun _{/\oLMod ^{\otimes }}(\pushout , D^{\otimes }) \] the full subcategory spanned by the enriched morphisms of \(D\), where \(\pushout \) is the walking span of Definition 1.4.1. There are two forgetful functors \[ (s,t)\colon \Ar ^{\enr }(D) \to D \times D \] given by \(s(\alpha ,\beta ):= N\) and \(t(\alpha ,\beta ) := M\).