A series of poly(${varepsilon}$-caprolactone)-grafted poly(${eta}$-amino ester)-poly(ethylene glycol)-poly(${eta}$-amino ester) ((PAE-g-PCL)-PEG-(PAE-g-PCL)) copolymers were designed and synthesized for use in injectable pH- and temperature-sensitive hydrogels. First, PEG diacrylate was synthesized and used as a macroinitiator, and the chain was extended by 4,4 trimethylene dipiperidine (TMDP), 3-amino-1-propanol and diacrylate monomers (1,4-butanediol diacrylate (BDA), and 1,6-hexanediol diacrylate (HDA)) to obtain the triblock copolymers (PAE-PEG-PAE). Poly (${varepsilon}$-caprolactone) was then grafted to the pendant OH groups of the resulting PAE-PEG-PAE by ring opening polymerization. The PAE-PEG-PAE and (PAE-g-PCL)-PEG-(PAE-g-PCL) copolymers were characterized by $^1H$ nuclear magnetic resonance (NMR) spectroscopy and gel permeation chromatography. The phase diagram was measured using the tube inverting method. At pH 6.6 or higher, the (PAE-g-PCL)-PEG-(PAE-g-PCL) copolymer solutions (30 wt%) displayed sol-gel-sol transitions with increasing temperature. The viscosity of the graft copolymer solutions was determined as a function of temperature by dynamic mechanical analysis. In addition, after the subcutaneous injection of the polymer solution (30 wt%, pH 6.4) into a rat, a gel formed within a short time, suggesting biomedical applications.