Sufficient/complete statistic $\leftrightarrow$ injective/surjective map?

I can't understand the paragraph in Completeness (statistics) - Wikipedia : We have an identifiable model space parameterised by $\theta$ , and a statistic $T$ . Then consider the map $f:p_{\theta }\mapsto p_{T|\theta }$ which takes each distribution on model parameter $\theta$ to its induced distribution on statistic $T$ . The statistic $T$ is said to be complete when $f$ is surjective, and sufficient when $f$ is injective. ( added on 2023-07-12 without any citation, and there's no revision after that) What does “ distribution on $\theta$ ” mean? Domain of $f$ is all prior distribution of $\theta$ (in Bayesian sense), or the famliy of distributions of samples ( $X_1, \ldots, X_n$ )? What is the codomain of $f$ ? I guess the image of $f$ is all possible distribution of $T$ (i.e. the famliy of distributions of $T$ ), but codomain should be larger than that, or $f$ is always surjective. My thoughts Sufficiency and completeness are related but independent concepts, as discussed in the following questions. If the statement in Wikipedia is true, then it’s a clear explanation of sufficiency and completeness. exponential family - Are complete statistics always sufficient? - Cross Validated Is a minimal sufficient statistic also a complete statistic - Cross Validated Roughly speaking: $T$ is sufficient : $T$ provides all information of $\theta$ from $X$ , and we can recover the whole distribution of $X$ once given $T$ . $f$ is injective : If we know that $y$ is $f$ of some $x$ , the…