Question about covariates (confounders)

Question

Assume this relationships with mediation: X --> Y --> Z now assume that covariate C influences the relationship X --> Y, Does this imply that we can use it in the relationship Y --> Z too?

in other word, if C is a covariate when X predicts Y. Is it also a covariate when Y predicts something else?

This happens many times in literature ... when it's known that some set of covariates are usually used to predict Y using X (as predictor). But then researcher direction could change, when he/she wants to predict some consequences of Y, lets call it Z-variable (so Y leads to Z). Now when putting our focus only on this new relationship (Y leads to Z), can we carry the previous set of covariates to use them in this relationship.

Answer 1

A variable $C$ is a confounder of the relationship between variables $A$ and $B$ if it causes $A$ and separately causes $B$. You said $C$ causes $X$ and causes $Y$, so it is a confounder of the $X \rightarrow Y$ relationship. You haven't said whether $C$ causes $Z$; if it does, then it is a confounder of the $Y \rightarrow Z$ relation; if not, it is not a confounder of that relationship. It is usually a safe assumption to make that if a variable affects prior steps in a causal chain, it may also affect letter steps; but there is a specific class of variables called "instrumental variables" for which this does not happen. Instrumental variables are very important in some specific forms of analysis.

Note: the word covariate does not mean confounder; it's any variable that you adjust for, correctly or incorrectly. See my answer here about this.