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set up a private image registry and push images to it

Yes, I just wanted to know if it is a easy way to avoid to push to a private registry

no. when you build a docker image it is just saved in the local docker image store. you cannot share that image store with your k3s node.

Yeah, basically we’d need to build a fake registry that accesses the local image storage and serves the image data.

For running k3s locally (or any local kubernetes variant), it's often easier to use ctlptl (https://github.com/tilt-dev/ctlptl). Ctlptl sets up a cluster using your chosen provider (k3d being one provider), and optionally launches a docker registry alongside.