Crossword puzzle clue "bull's eye or a hint to this puzzle's theme"

Bull's-eye ... or a hint to this puzzle's theme

Hint (and a hint to this puzzle's theme)

Keeps an eye on, and a hint to this puzzle's theme

Puzzle invented by Lewis Carroll (and a hint to this puzzle's theme)

Eye the bull's-eye

Eye, as a bull's-eye

Contented sighs (and a homophonic hint to this puzzle's theme)

"Dog the Bounty Hunter" network, and a hint to the theme in this puzzle's four longest answers