From p.256 to p.259, Griffiths and Harris verify Riemann's count explicitly in cases g=3,4 and 5. In each case of $g=3,4,5$, one important step is to claim that curves $C\subseteq\mathbb P^{g-1}$ satisfying certain conditions are canonical curves.
Here are the statements. In case $g=3$
(p.256-p.258) ... and conversely if $C\subseteq \mathbb P^2$ is a smooth quartic curve, by the adjunction formula, $K_C=\cdots=H|_C$, so $C$ is a canonical curve.
and in case $g=4$ on p.258
Next, let $C\subseteq\mathbb P^3$ be a canonical curve of genus $4$. ... ... Conversely, by the adjunction formula, for any cubic $Q'$ and quadric $Q$ meeting in a smooth curve $C$, we have $K_{Q'}=\cdots=-H_{Q'}$ and $K_C=\cdots=H|_C$, so $C$ is a canonical curve of genus $4$.
and in case $g=5$ on p.259
For $C\subseteq\mathbb P^4$ a canonical curve of genus $5$, ... ... Conversely, if $Q$, $Q'$ and $Q''$ are any three quadrics in $\mathbb P^4$ meeting transversely, the adjunction formula applied three times tells us that $C$ is a canonical curve of genus $5$.
These areguments all applied the adjunction formula to claim $K_C=H|_C$. However it is not true in general to claim $C$ is a canonical curve only from $K_C=H|_C$. Can someone tell me what is the hidden theorem used here?
If $C\subset\mathbb{P}^n=P$ is a smooth projective curve, one needs two conditions to make it a canonical embedding. $K_C=O_P(1)_{|C}$ and the natural map $H^0(O_P(1))\to H^0(K_C)$ is an isomorphism.
In your case, the curves in question are complete intersections and non- degenerate and thus both the conditions will be satisfied.