TIG's two suggestions are excellent. I will add that if you've applied the material to the component 'wrapper' instead of the faces, you won't have the option to move the texture as in his second suggestion. My preference is to always apply the materials to faces and not the component 'wrapper' so that I have the option to adjust the material without changing the component axes.
As to the location of the component axes, you may want to move them to make inserting instances easier. This would be especially true if you expect to reuse the component in other models later.