Spahn factorization of a monad

Example

Let $(i_!\dashv i^*\dashv i_*\dashv i^!):H_{th}\stackrel{\i^!}{\to} H$ be an adjoint quadruple defining infinitesimal cohesion; i.e. $i_!$ and $i_*$ are full and faithful and $i_!$ preserves the terminal object.

Then there are two factorizations of the terminal geometric morphism? on $H_{th}$ through the geometric morphisms $(i^*\dashv i_*)$ and $(i_*\dashv i^!)$:

$$\array{ \mathbf{H} && \stackrel{(i^* \dashv i_*)}{\to} && \mathbf{H}_{th} \\ & {}_{\mathllap{\Gamma_H}}\searrow && \swarrow_{\mathrlap{\Gamma_{th}}} \\ && \infty Grpd }$$

as is the induced geometric morphism $(i_* \dashv i^!) : \mathbf{H}_{th} \to \mathbf{H}$

$$\array{ \mathbf{H}_{th} && \stackrel{(i_* \dashv i^!)}{\to} && \mathbf{H} \\ & {}_{\mathllap{\Gamma_{th}}}\searrow && \swarrow_{\mathrlap{\Gamma_H}} \\ && \infty Grpd } \,.$$