Within a sequence-to-sequence decoder, the RNN updates its hidden state at each time step $t'$ using a transformation function $g$ that takes three inputs: the previous target token $y_{t'-1}$, the encoder's context variable $\mathbf{c}$, and the decoder's hidden state from the preceding time step $\mathbf{s}_{t'-1}$. The resulting hidden state update is expressed as:

$\mathbf{s}_{t'} = g(y_{t'-1}, \mathbf{c}, \mathbf{s}_{t'-1})$

This formulation mirrors the encoder's recurrence but differs in a key way: the decoder's transformation incorporates the context variable $\mathbf{c}$ as an additional input at every time step, ensuring that the encoded source sequence information continuously influences the generation process.