Check the forward propagation dimensions for the hidden layer
\[\begin{equation*}
\boxed{A^{[2]}_{p^{[2]} \times n^{[2]}}}
\stackrel{?}{=}
\boxed{A^{[2]}_{p^{[2]} \times p^{[1]}}}
\end{equation*}
\]
with
\[\begin{equation*}
p^{[1]} = n^{[1]}
\end{equation*}
\]
by
\[\begin{align*}
\boxed{A^{[2]}_{p^{[2]} \times n^{[2]}}}
& =
g^{[2]}
\big(
W^{[2]}_{p^{[2]} \times n^{[2]}}
A^{[1]}_{p^{[1]} \times n^{[1]}}
+ \vec{b}^{[2]}_{p^{[2]} \times 1}
\big)
&& \text{by definition of \(A^{[i]}\)}
\\
& =
g^{[2]}
\big(
W^{[2]}_{p^{[2]} \times p^{[1]}}
A^{[1]}_{p^{[1]} \times n^{[1]}}
+ \vec{b}^{[2]}_{p^{[2]} \times 1}
\big)
&& \text{substitute \(n^{[2]}\) with \(p^{[1]}\)}
\\
& =
g^{[2]}
\big(
T^{[2]}_{p^{[2]} \times n^{[1]}}
+ \vec{b}^{[2]}_{p^{[2]} \times 1}
\big)
&& \text{multiply \(W^{[2]} A^{[1]}\)}
\\
& =
g^{[2]}
\big(
U^{[2]}_{p^{[2]} \times n^{[1]}}
\big)
&& \text{add \(T^{[2]} + \vec{b}^{[2]}\)}
\\
& =
g^{[2]}
\big(
U^{[2]}_{p^{[2]} \times p^{[1]}}
\big)
&& \text{substitute \(n^{[1]}\) with \(p^{[1]}\)}
\\
& =
\boxed{A^{[2]}_{p^{[2]} \times p^{[1]}}}
&& \text{apply \(g^{[2]}\).}
\end{align*}
\]