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powering of ∞-toposes over ∞-groupoids – section

$$\begin{array}{lll} & \mathbf{H} \bigg( (-) ,\, Maps \Big( \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) \bigg) \\ & \;\simeq\; \mathbf{H} \Big( (-) \times \underset{\longrightarrow}{\lim} \, LConst(S_\bullet) ,\, X \Big) & \text{ (1) } \\ & \;\simeq\; \mathbf{H} \Big( \underset{\longrightarrow}{\lim} \big( (-) \times LConst(S_\bullet) \big) ,\, X \Big) & \text{ [[universal colimits]] } \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \big( (-) \times LConst(S_\bullet) ,\, X \big) \\ & \;\simeq\; \underset{\longleftarrow}{\lim} \, \mathbf{H} \Big( (-) ,\, Maps \big( LConst(S_\bullet) ,\, X \big) Big) \\ & \;\simeq\; \mathbf{H} \Big( (-) ,\, \underset{\longleftarrow}{\lim} \, Maps \big( LConst(S_\bullet) ,\, X \big) Big) \end{array}$$

(1)

$$\begin{array}{lll} & (1) \\ & \text{(1)} \\ & \text{ <a href="https://ncatlab.org/nlab/show/terminal+geometric+morphism#DirectImageOfTerminalGeometricMoprhismIsHomOutOfTerminalObject">here</a>} \end{array}$$

For the first statement:

$$\begin{array}{lll} \mathbf{H} \Big( X ,\, Maps \big( LConst(S) ,\, A \big) \Big) & \;\simeq\; \mathbf{H} \big( X \times LConst(S) ,\, A big) \\ & \;\simeq\; \mathbf{H} \Big( LConst(S) ,\, Maps \big( X ,\, A \big) \Big) \\ & \;\simeq\; Grpd_\infy \Big( S ,\, \Gamma \, Maps \big( X ,\, A \big) \Big) \\ & \;\simeq\; Grpd_\infy \Big( S ,\, \mathbf{H} \big( X ,\, A \big) \Big) \end{array}$$