terminal geometric morphism in nLab

Proof

Since every ∞ \infty -groupoid is an ∞ \infty -colimit (over itself) of the point (see there):

(2)S≃lim⟶S* S \,\simeq\, \underset{\underset{S}{\longrightarrow}}{\lim} \,\ast

and since the inverse image of a geometric morphism preserves finite limits (by definition), such as the terminal object, and all ∞ \infty -colimits (since left adjoints preserve colimits), we have that LConstLConst must be given by forming the corresponding ∞\infty-colimit of copies of the terminal object * H\ast_{\mathbf{H}} in H\mathbf{H}, which does exist:

LConst(S) ≃LConst(lim⟶S*) ≃lim⟶SLConst(*) ≃lim⟶S* H \begin{aligned} LConst(S) & \;\simeq\; LConst \Big( \underset{ \underset{S}{\longrightarrow} }{\lim} \, \ast \Big) \\ & \;\simeq\; \underset{ \underset{S}{\longrightarrow} }{\lim} \, LConst \big( \ast \big) \\ & \;\simeq\; \underset{ \underset{S}{\longrightarrow} }{\lim} \, \ast_{\mathbf{H}} \end{aligned}

Proposition

The direct image of the terminal geometric morphism (1) is given by the hom-space out of the terminal object, in that for X∈HX \,\in\, \mathbf{H} there is a natural equivalence

Γ(X)≃H(* H,X), \Gamma (X) \;\simeq\; \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \,,

where * H∈H\ast_{\mathbf{H}} \,\in\, \mathbf{H} denotes the terminal object.

Proof

For all S∈Grpd ∞S \,\in\, Grpd_\infty we have the following sequence of natural equivalences:

Grpd ∞(S,H(* H,X)) ≃Grpd ∞(lim⟶S*,H(* H,X)) ≃lim⟵SGrpd ∞(*,H(* H,X)) ≃lim⟵SH(* H,X) ≃H(lim⟵S* H,X) ≃H(lim⟵SLConst(*),X) ≃H(LConst(lim⟵S*),X) ≃H(LConst(S),X) \begin{array}{lll} Grpd_\infty \big( S ,\, \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \big) & \;\simeq\; Grpd_\infty \Big( \underset{\underset{S}{\longrightarrow}}{\lim} \ast ,\, \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \Big) \\ & \;\simeq\; \underset{\underset{S}{\longleftarrow}}{\lim} \, Grpd_\infty \big( \ast ,\, \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \big) \\ & \;\simeq\; \underset{\underset{S}{\longleftarrow}}{\lim} \, \mathbf{H}(\ast_{\mathbf{H}} ,\, X) \\ & \;\simeq\; \mathbf{H} \Big( \underset{\underset{S}{\longleftarrow}}{\lim} \ast_{\mathbf{H}} ,\, X \Big) \\ & \;\simeq\; \mathbf{H} \Big( \underset{\underset{S}{\longleftarrow}}{\lim} \, LConst(\ast) ,\, X \Big) \\ & \;\simeq\; \mathbf{H} \Big( LConst \big( \underset{\underset{S}{\longleftarrow}}{\lim} \ast \big) ,\, X \Big) \\ & \;\simeq\; \mathbf{H} \big( LConst(S) ,\, X \big) \end{array}

(Here we used (2) and that hom-functor preserves limits and that left adjoints preserve colimits and that LConstLConst preserves finite limits such as the terminal object, by definition.)

But this is a hom-equivalence which exhibits (see here) H(*,−)\mathbf{H}(\ast,-) as a right adjoint ∞ \infty -functor to LConstLConst. This implies the claim by essential uniqueness of adjoints.