Let $E$ be a set, $\mathcal{E}$ a $\sigma$-algebra on $E$ and $\mu : \mathcal{E} \to [0, \infty]$ a measure on $\mathcal{E}$.
Set $\mathcal{N}^\mu := \{ A \subseteq E \mid \exists B \in \mathcal{E}, A \subseteq B, \mu(B) = 0 \}$ as the collection of subsets of $\mu$-null sets. Then $\mathcal{E}^\mu := \sigma(\mathcal{E} \cup \mathcal{N}^\mu)$ is the completion of $\mathcal{E}$: it is the smallest $\mu$-complete $\sigma$-algebra larger than $\mathcal{E}$.
Let $\mathcal{F} \subseteq \mathcal{E}$ be a sub-$\sigma$-algebra. Then the restriction $\mu|_{\mathcal{F}} : \mathcal{F} \to [0, \infty]$ is a measure on $\mathcal{F}$. Hence we can consider the $\mu|_\mathcal{F}$-completion $\mathcal{F}^{\mu|_\mathcal{F}} = \sigma(\mathcal{F} \cup \mathcal{N}^{\mu|_\mathcal{F}})$ of $\mathcal{F}$.
I often see in books on stochastic processes also the following definition: $\mathcal{F}^\mu := \sigma(\mathcal{F} \cup \mathcal{N}^\mu)$. The difference is that in $\mathcal{F}^\mu$ we add in all (subsets of) $\mu$-null sets of $\mathcal{E}$ (and not only those of $\mathcal{F}$).
It holds: $\mathcal{F}^{\mu|_\mathcal{F}} \subseteq \mathcal{F}^\mu \subseteq \mathcal{E}^\mu$ and all the inclusions may be strict. For an extreme case, just consider $\mathcal{F}$ the trivial $\sigma$-algebra.
What is the canonical or natural definition of a $\mu$-completion of a sub-$\sigma$-algebra - especially in the context of stochastic processes and completions of filtrations? (It would also be interesting to have an aswer in categorical terms, explaining the naturality of the construction.)